URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTENCE OF WEAK SOLUTIONS FOR NONLINEAR SYSTEMS INVOLVING SEVERAL P-LAPLACIAN OPERATORS SALAH A

Electronic Journal of Differential Equations, Vol. 2009(2009), No. 81, pp. 1–10.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu

EXISTENCE OF WEAK SOLUTIONS FOR NONLINEAR SYSTEMS INVOLVING SEVERAL P-LAPLACIAN OPERATORS

SALAH A. KHAFAGY, HASSAN M. SERAG

Abstract. In this article, we study nonlinear systems involving several p- Laplacian operators with variable coefficients. We consider the system

−∆p_iui=aii(x)|u_i|^pi−2ui−

n

X

j6=i

aij(x)|u_i|^αi|u_j|^αjuj+fi(x),

where ∆p denotes thep-Laplacian defined by ∆pu ≡ div[|∇u|^p−2∇u] with p > 1, p 6= 2; αi ≥ 0; fi are given functions; and the coefficients aij(x) (1≤i, j≤n) are bounded smooth positive functions. We prove the existence of weak solutions defined on bounded and unbounded domains using the theory of nonlinear monotone operators.

1. Introduction

The generalized formulation of many boundary-value problems for partial differ- ential equations leads to operator equations of the form

A(u) =f

on a Banach space V. For this operator equation, we have the so-called weak formulation:

Findu∈V such that (A(u), v) = (f, v) for all v∈V.

Then functional analysis has tools for proving existence of generalized (weak) solutions for a relatively wide class of differential equations that appear in mathe- matical physics and industry.

The existence of weak solutions for 2×2 nonlinear systems involving severalp- Laplacian operators have been proved, using the method of sub and super solutions in [5], and using the theory of nonlinear monotone operators in [6].

Here, we use the theory of nonlinear monotone operators to prove the existence of weak solutions for the following nonlinear systems involving severalp-Laplacian operators with variable coefficients defined on a bounded domain Ω of R^N with

2000Mathematics Subject Classification. 74H20, 35J65.

Key words and phrases. Existence of weak solution; nonlinear system, p-Laplacian.

c

2009 Texas State University - San Marcos.

Submitted December 9, 2008. Published July 10, 2009.

1

boundary∂Ω,

−∆p_iui≡ −div[|∇ui|^pi−2∇ui]

=aii(x)|ui|^pi−2ui−

n

X

j6=i

aij(x)|ui|^αi|uj|^αjuj+fi(x) in Ω, ui= 0, i= 1,2, . . . , n, on∂Ω.

Then, we generalize our results to systems defined on the whole spaceR^N. This article is organized as follow: In section 2 we introduce some technical results and definitions concerning the theory of nonlinear monotone operators. We study the existence of weak solutions for n×n nonlinear systems defined on a bounded domain in section 3, and on unbounded domains in section 4.

2. Preliminary results

First, we introduce some results concerning the theory of nonlinear monotone operators [4].

LetA:V →V⁰be an operator on a Banach spaceV. We say that the operator Ais:

Bounded if it maps bounded sets into bounded; i.e., for each r > 0 there exists M >0 (M depending onr) such that

kuk ≤rimplieskA(u)k ≤M, ∀u∈V; coercive if lim_kuk→∞hA(u), ui/kuk=∞;

monotone ifhA(u1)−A(u2), u1−u2i ≥0 for allu1, u2∈V;

strictly monotone ifhA(u1)−A(u2), u1−u2i>0 for allu1, u2∈V,u16=u2; continuous ifuk →uimpliesA(uk)→A(u), for all uk, u∈V;

strongly continuous ifu_k →^w uimpliesA(u_k)→A(u), for allu_k, u∈V;

continuous on finite-dimensional subspaces if A: V_n → V_n⁰ is continuous for each subspaceV_n of finite dimension.

demicontinuous ifu_k→uimpliesA(u_k)→^w A(u), for allu_k, u∈V; the operatorA is said to be satisfy the M0-condition ifuk

→w u, A(uk)→^w f, and [hA(uk), uki → hf, ui] implyA(u) =f.

Remark 2.1. (i) Strongly continuous operators are continuous, and they are continuous on finite dimensional subspaces.

(ii) Strongly continuous operators are bounded and satisfy theM0-condition.

(iii) Strictly monotone operators are monotone operators.

(iv) Monotone and continuous operators satisfy theM0-condition.

Theorem 2.2. Let V be a separable reflexive Banach space and A :V → V⁰ an operator which is: coercive, bounded, continuous on finite-dimensional subspaces and satisfying the M₀−condition. Then the equation A(u) = f admits a solution for eachf ∈V⁰.

Next, we introduce the Sobolev space W^1,p(Ω),1 < p < ∞, defined as the completion ofC^∞(Ω) with respect to the norm (see [1])

kuk_W1,p =hZ

Ω

|∇u|^p+|u|^pi^1/p

<∞. (2.1)

Since we are studying a Dirichlet problem, we define the space W₀^1,p(Ω) as the closure ofC₀^∞(Ω) inW^1,p(Ω) with respect to the norm

kuk_W1,p 0 =hZ

Ω

|∇u|^pi^1/p

<∞, (2.2)

which is equivalent to the norm given by (2.1). Both spacesW^1,p(Ω) andW₀^1,p(Ω) are well defined reflexive Banach Spaces. The spaceW₀^1,p(Ω) is compactly imbedded in the spaceL^p(Ω); i.e.,

W₀^1,p(Ω),→,→L^p(Ω), (2.3) which implies

kukL^p(Ω)≤ckuk_W^1,p

0 (Ω), i.e., Z

Ω

a(x)|u|^p≤c⁰ Z

Ω

|∇u|^p (2.4) for everyu∈W₀^1,p(Ω), wherea(x) is a smooth bounded positive function.

Now, we introduce some results [2] concerning the eigenvalue problem

−∆pu≡ −div[|∇u|^p−2∇u] =λa(x)|u|^p−2u in Ω,

u= 0 on∂Ω. (2.5)

We will say that λ ∈ R is an eigenvalue of (2.5) if there exists u ∈ W₀^1,p(Ω), u6= 0, such that

Z

Ω

|∇u|^p−2∇u∇ϕ=λ Z

Ω

a(x)|u|^p−2uϕ

hods for allϕ∈W₀^1,p(Ω). Thenuis called an eigenfunction corresponding to the eigenvalueλ.

Lemma 2.3. The eigenvalue problem (2.5) admits a positive principal eigenvalue λ=λa(Ω) >0 which is associated with a positive eigenfunction u≥0 a.e. in Ω normalized bykukp= 1. Moreover, the first eigenvalue is characterized by

λa(Ω) = inf Z

Ω

|∇u|^p: Z

Ω

a(x)|u|^p= 1 . (2.6)

Also, from the characterization of the first eigenvalue given by (2.6), we have λ_a(Ω)

Z

Ω

a(x)|u|^p≤ Z

Ω

|∇u|^p. (2.7)

3. Nonlinear systems defined on bounded domains Let us consider the nonlinear system

−∆p_iui=aii(x)|ui|^pi−2ui−

n

X

j6=i

aij(x)|ui|^αi|uj|^αjuj+fi(x) in Ω, ui= 0, i= 1,2, . . . , n, on∂Ω,

(3.1)

whereaii(x) is a smooth bounded positive function, Ω is a bounded domain ofR^N, and

α_i≥0, f_i∈L^p∗i(Ω), (3.2)

1 p_i + 1

p^∗_i = 1, αi+ 1 p_i = 1

2, i= 1,2, . . . , n. (3.3)

Theorem 3.1. For (fi) ∈ Qn

i=1L^p∗i(Ω), there exists a weak solution (ui) in the spaceQn

i=1W₀^1,pi(Ω) for the system (3.1), if

λ_aii(Ω)>1, i= 1,2, . . . , n. (3.4) Proof. We transform the weak formulation of (3.1) to the operator form (A−B)U = F, where,A, B andF are operators defined on Qn

i=1W₀^1,pi(Ω) by (AU,Φ)≡(A(u1, u2, . . . , un),(φ1, φ2, . . . , φn)) =

n

X

i=1

Z

Ω

|∇ui|^pi−2∇ui∇φi, (3.5) (BU,Φ)≡(B(u₁, u₂, . . . , u_n),(φ₁, φ₂, . . . , φ_n))

=

n

X

i=1

[ Z

Ω

aii(x)|ui|^pi−2uiφi−

n

X

j6=i

Z

Ω

aij(x)|ui|^αi|uj|^αjujφi], (3.6)

(F,Φ)≡((f₁, f₂, . . . , f_n),(φ₁, φ₂, . . . , φ_n)) =

n

X

i=1

Z

Ω

f_iφ_i. (3.7)

Now, consider the operatorJ defined by (J(u), φ) =

Z

Ω

|∇u|^p−2∇u∇φ. (3.8)

This operator is bounded: Since

|(J(u), φ)| ≤ Z

Ω

|∇u|^p−1|∇φ|,

using H¨older’s inequality, we obtain

|(J(u), φ)| ≤hZ

Ω

|∇u|^pi^p−1_p hZ

Ω

|∇φ|^pi^1/p

=kuk^p−1

W₀^1,p(Ω)kφk_W^1,p

0 (Ω). Also, we can prove thatJ is continuous, let us assume that u_k →u in W₀^1,p(Ω).

Then ku_k−uk_W1,p

0 (Ω)→ 0, so thatk∇u_k− ∇uk_Lp(Ω)→ 0. Applying Dominated Convergence Theorem, we obtain

k(|∇uk|^p−2∇uk− |∇u|^p−2∇u)k_Lp(Ω)→0, and hence

kJ(u_k)−J(u)k_Lp(Ω) ≤ k(|∇uk|^p−2∇uk− |∇u|^p−2∇u)kL^p(Ω)→0.

Finally,J is strictly monotone:

(J(u1)−J(u2), u1−u2) = Z

Ω

|∇u1|^p−2∇u1∇u1+ Z

Ω

|∇u2|^p−2∇u2∇u2

− Z

Ω

|∇u1|^p−2∇u1∇u2− Z

Ω

|∇u2|^p−2∇u2∇u1;

using H¨older’s inequality, we obtain (J(u1)−J(u2), u1−u2)

≥ Z

Ω

|∇u₁|^p+ Z

Ω

|∇u₂|^p−hZ

Ω

|∇u₁|^pi^p−1_p hZ

Ω

|∇u₂|^pi¹_p

−hZ

Ω

|∇u₂|^pi^p−1_p hZ

Ω

|∇u₁|^pi1/p

=ku1k^p

W₀^1,p(Ω)+ku2k^p

W₀^1,p(Ω)− ku1k^p−1

W₀^1,p(Ω)ku2k_W1,p

0 (Ω)− ku2k^p−1

W₀^1,p(Ω)ku1k_W1,p 0 (Ω), and hence,

(J(u₁)−J(u₂), u₁−u₂)

≥(ku1k^p−1

W₀^1,p(Ω)− ku2k^p−1

W₀^1,p(Ω))(ku1k_W^1,p

0 (Ω)− ku2k_W^1,p

0 (Ω))>0.

Now,AU can be written as the sum ofJ1(u1), J2(u2), . . . , Jn(un) where (Ji(ui), φi) =

Z

Ω

|∇ui|^pi−2∇ui∇φi, i= 1,2, . . . , n,

and as above, the operatorsJ₁,J₂, . . . andJ_n are bounded, continuous and strictly monotone; so their sum, the operatorA, will be the same.

For the operatorB, B:

n

Y

i=1

W₀^1,pi(Ω)→

n

Y

i=1

L^pi(Ω),

we can prove that it is a strongly continuous operator. To prove that, let us assume that uik

→w ui in W₀^1,pi(Ω), i = 1,2, . . . , n. Then, using (2.3), (uik) → (ui) in Qn

i=1L^pi(Ω). By the Dominated Convergence Theorem,

aii(x)|uik|^pi−2uik→aii(x)|ui|^pi−2ui in L^pi(Ω),

−aij(x)|uik|^αi|ujk|^αjujk→ −aij(x)|u_i|^αi|uj|^αjuj in L^pj(Ω), Since

(BU_k−BU, W) = (B(u_1k, u_2k, . . . , u_nk)−B(u₁, u₂, . . . , u_n),(w₁, w₂, . . . , w_n))

=

n

X

i=1

hZ

Ω

a_ii(x)(|u_ik|^pi−2u_ik− |u_i|^pi−2u_i)w_i

−

n

X

j6=i

Z

Ω

aij(x)(|uik|^αi|ujk|^αjujk− |u_i|^αi|uj|^αjuj)wi

i , it follows that

kBUk−BUk ≤

n

X

i=1

hkaii(x)(|uik|^pi−2uik− |ui|^pi−2ui)k_Lpi(Ω)

+

n

X

j6=i

kaij(x)(|uik|^αi|ujk|^αj+1− |u_i|^αi|uj|^αj+1)

k_Lpi(Ω)]→0.

This proves thatB is a strongly continuous operators. According to Remark 2.1, the operator A−B satisfies the M₀-condition. Now, to apply Theorem 2.2, it

remains to prove thatA−B is a coercive operator ((A−B)U, U)

=

n

X

i=1

Z

Ω

|∇ui|^pi−

n

X

i=1

hZ

Ω

aii(x)|ui|^pi−

n

X

j6=i

Z

Ω

aij(x)|u_i|^αi+1|uj|^αj+1i

≥

n

X

i=1

Z

Ω

|∇ui|^pi−

n

X

i=1

Z

Ω

aii(x)|ui|^pi. Using (2.7), we obtain

((A−B)U, U)≥

n

X

i=1

Z

Ω

|∇u_i|^pi−

n

X

i=1

1 λaii(Ω)

Z

Ω

| 5u_i|^pi

=

n

X

i=1

(1− 1 λ_aii(Ω))

Z

Ω

| 5ui|^pi, and hence,

((A−B)U, U)≥k

n

X

i=1

kuik^pi

W₀^1,pi(Ω)=kk(ui)kQn

i=1W₀^1,pi(Ω). So that

((A−B)U, U)→ ∞ ask(u_i)kQn

i=1W₀^1,pi(Ω)→ ∞.

This proves the coercivity condition and so, the existence of a weak solution for

systems (3.1).

4. Nonlinear systems defined on R^N We consider the nonlinear system

−∆piu_i=a_ii(x)|ui|^pi−2u_i−

n

X

j6=i

a_ij(x)|ui|^αi|uj|^αju_j+f_i(x), x∈R^N, lim

|x|→∞u_i(x) = 0, i= 1,2, . . . , n, x∈R^N.

(4.1)

We assume that 1< pi < N,i= 1,2, . . . , n, and the coefficientsaii(x) andaij(x) are smooth bounded positive functions such that

0< aii(x)∈L^piN(R^N)∩L^∞(R^N), 0< aij(x)∈L

N

αi+αj+2(R^N)∩L^∞(R^N). (4.2) To discuss this problem, we need the following results which are studied in [3] and that we recall briefly.

Let us introduce the Sobolev reflexive Banach space

D^1,p(R^N) ={u∈L^{N−pN p} (R^N) :∇u∈(L^p(R^N))ⁿ}, which is defined as the completion ofC₀^∞(R^N) with respect to the norm

kuk_D1,p(R^N)=hZ

R^N

|∇u|^pi1/p

<∞. (4.3)

Moreover D^1,p(R^N) is embedded continuously in the space L^{N−pN p} (R^N); that is, D^1,p(R^N),→L^{N−pN p} (R^N), which implies

kuk

L

N p

N−p(R^N)≤k kukD^1,p(R^N). (4.4)

Lemma 4.1. The eigenvalue problem

−∆_Pu≡ −div[|∇u|^p−2∇u] =λa(x)|u|^p−2u in R^N,

u(x)→0 as|x| → ∞, u >0 inR^N, (4.5) admits a positive principal eigenvalueλ=λ_a(Ω)which is associated with a positive eigenfunction u∈D^1,p(R^N). Moreover, the principal eigenvalue λa(Ω) is charac- terized by

λa(Ω) Z

R^N

a(x)|u|^p≤ Z

R^N

|∇u|^p, ∀ u∈D^1,p(R^N) (4.6) where

0< a(x)∈L^Np(R^N)∩L^∞(R^N). (4.7) In this section, we assume that

αi ≥0, fi∈L^{N(piN pi−1)+pi}(R^N), αi+αj+ 2< N,1< pi< n 1

pi

+ 1

p^∗_i = 1, α_i+ 1 pi

= 1

2, i= 1,2, . . . , n.

(4.8)

Theorem 4.2. For(f_i)∈Qn

i=1L^{N(piN pi−1)+pi}(R^N), there exists a weak solution (u_i) inQn

i=1D^1,pi(R^N)for system (4.1), if

λa_ii(Ω)>1, i= 1,2, . . . , n. (4.9) Proof. As in section 3, we transform the weak formulation of the system (4.1) to the operator form (A−B)U = F, where, A, B and F are operators defined on Qn

i=1D^1,pi(R^N) by

(AU,Φ)≡(A(u₁, u₂, . . . , u_n),(φ₁, φ₂, . . . , φ_n))

=

n

X

i=1

Z

R^N

|∇ui|^pi−2∇ui∇φi =

n

X

i=1

(Ji(ui), φi), (4.10) (BU,Φ)≡(B(u₁, u₂, . . . , u_n),(φ₁, φ₂, . . . , φ_n))

=

n

X

i=1

[ Z

R^N

a_ii(x)|u_i|^pi−2u_iφ_i−

n

X

j6=i

Z

R^N

a_ij(x)|u_i|^αi|u_j|^αju_jφ_i], (4.11)

(F,Φ)≡((f₁, f₂, . . . , f_n),(φ₁, φ₂, . . . , φ_n)) =

n

X

i=1

Z

R^N

f_iφ_i. (4.12) First, we prove thatA, B andF are bounded operators onQn

i=1D^1,pi(R^N).

For the operatorA, by using (4.10) and applying Holder inequality, we have

|(AU,Φ)| ≤

n

X

i=1

Z

R^N

|∇ui|^pi−1|∇φi|

≤

n

X

i=1

hZ

R^N

|∇ui|^pii^(pi−1)/pihZ

R^N

|∇φi|^pii^1/pi

=

n

X

i=1

kuik^p_Dⁱ1⁻¹,pi(R^N)kφik_D¹,pi(R^N)

=Xⁿ

i=1

ku_ik^p_Dⁱ₁⁻¹_,pi(

R^N)

k(φ_i)k^Qn

i=1D^1,pi(R^N)

.

This proves the boundedness of the operatorA.

For the operatorB, we have

|(BU,Φ)| ≤

n

X

i=1

[ Z

R^N

aii(x)|ui|^pi−1|φi|+

n

X

j6=i

Z

R^N

aij(x)|ui|^αi|uj|^αj+1|φi|]

≤

n

X

i=1

Z

R^N

aii(x)^Np_N^pZ

R^N

|ui|^{N−N pipi(pi}

−1)(N−pi)

N pi Z

R^N

|φi|^{N−N pipi}

N−pi N pi

+

n

X

j6=i

hZ

R^N

(aij(x))

N αi+αj+2i^αi

+αj+2

N hZ

R^N

|ui|^{N−N pipi}i^αi

(N−pi) N pi

×hZ

R^N

|uj|

N pj N−pji

(αj+1)(N−pj)

N pj hZ

R^N

|φi|^{N−N pipi}i

N−pi N pi

≤

n

X

i=1

hk_ikuik^p_Dⁱ1⁻¹,pi(R^N)kφik_D¹,pi(R^N)

+

n

X

j6=i

l_iku_ik^α_Dⁱ1,pi(R^N)ku_jk^αj+1

D^1,pj(R^N)kφ_ik_D1,pi(R^N)

i

=hXⁿ

i=1

hk_ikuik^p_Dⁱ⁻¹1,pi(R^N)+

n

X

j6=i

l_ikuik^α_Dⁱ1,pi(R^N)kujk^α_D^j1,pj⁺¹(R^N)

ii

× k(φi)k^Qn

i=1D¹,pi(R^N)

For the operatorF, we have (F,Φ) =PN i=1

R

Rⁿfiφi and so

|(F,Φ)|=

N

X

i=1

Z

Rⁿ

fiφi

≤

N

X

i=1

hZ

Rⁿ

|f_i|^{n(pinpi−1)+pi}i

n(pi−1)+pi

npi hZ

Rⁿ

|φ_i|^n−npipii

n−pi npi

=

N

X

i=1

(kfik

L

npi n(pi−1)+pi(Rⁿ)

)k(φi)k^QN

i=1D¹,pi(Rⁿ). Now, as in section 3, the operator A defined by (AU,Φ) = Pn

i=1(J_i(u_i),Φ) is continuous. Also it is strictly monotone onQn

i=1D^1,pi(R^N), since (Ji(u1)−Ji(u2), u1−u2)

≥(ku1k^p_Dⁱ1⁻¹,pi(R^N)− ku2k^p_Dⁱ⁻¹1,pi(R^N))(ku1k_D1,pi(R^N)− ku2k_D1,pi(R^N))>0.

For the operatorB, we can prove that it is a strongly continuous operator by using Dominated Convergence theorem and continuous imbedding property for the space Qn

i=1D^1,pi(R^N) into Qn

i=1L^{N−N pipi}(R^N). To prove that, let us assume that u_ik →^w u_i in D^1,pi(R^N), i = 1,2, . . . , n. Then (u_ik) → (u_i) in Qn

i=1L^{N−N pipi}(R^N).

Now, the sequence (uik) is bounded inD^1,pi(R^N),i= 1,2, . . . , n, then it is contain- ing a subsequence again denoted by (u_ik) converges strongly to u_i in L^{N−N pipi}(B_r0), i = 1,2, . . . , n, for any bounded ball B_r0 = {x ∈ R^N : kxk ≤ r₀}. Since

u_ik, u_i∈L^{N−N pipi}(B_r0), Then using the Dominated Convergence Theorem, we have ka_ii(x)(|u_ik|^pi−2u_ik− |u_i|^pi−2u_i)k N pi

N(pi−1)+pi

→0, kaij(x)(|uik|^αi−1|ujk|^αj+1ujk− |ui|^αi−1|uj|^αj+1uj)k N pi

N(pi−1)+pi

→0, fori= 1,2, . . . , n. Since

((BUk−BU), W) = (B(u1k, u2k, . . . , unk)−B(u1, u2, . . . , un),(w1, w2, . . . , wn))

=

n

X

i=1

hZ

R^N

a_ii(x)(|uik|^pi−2u_ik− |ui|^pi−2u_i)w_i

−

n

X

j6=i

Z

R^N

a_ij(x)(|u_ik|^αi|u_jk|^αju_jk− |u_i|^αi|u_j|^αju_j)w_ii , it follows that

kBU_k−BUk^Qn

i=1D^1,pi(Br0)

≤

n

X

i=1

hkaii(x)(|uik|^pi−2uik− |ui|^pi−2ui)k N pi N(pi−1)+pi

+

n

X

j6=i

kaij(x)(|uik|^αi|ujk|^αj+1− |u_i|^αi|uj|^αj+1)k N pi N(pi−1)+pi

i→0.

As in [6], we can prove that, the norm kBUk−BUk^Qn

i=1D^1,pi(R^N)

tends strongly to zero and then the operatorBis strongly continuous. According to Remark 2.1, the operatorA−Bsatisfies theM0-condition. Now, to apply Theorem 2.2, it remains to prove that the operatorA−B is a coercive operator,

((A−B)U, U)

=

n

X

i=1

Z

R^N

|∇ui|^pi−

n

X

i=1

hZ

R^N

aii(x)|ui|^pi−

n

X

j6=i

Z

R^N

aij(x)|u_i|^αi+1|uj|^αj+1i

≥

n

X

i=1

Z

R^N

|∇u_i|^pi−

n

X

i=1

Z

R^N

a_ii(x)|u_i|^pi. Using (4.6), we obtain

((A−B)U, U)≥

n

X

i=1

Z

R^N

|∇ui|^pi−

n

X

i=1

1 λa_ii(Ω)

Z

R^N

| 5ui|^pi

=

n

X

i=1

(1− 1 λa_ii(Ω))

Z

R^N

| 5u_i|^pi. From (4.9), we deduce

((A−B)U, U)≥k

n

X

i=1

kuik^p_Dⁱ1,pi(R^N)=kk(ui)k^Qn

i=1D¹,pi(R^N). So that ((A−B)U, U)→ ∞ask(ui)k^Qn

i=1D^1,pi(R^N)→ ∞. This proves the coercivity condition and so, the existence of a weak solution for systems (4.1).

Acknowledgments. The authors wish to thank the anonymous referees for their interesting remarks.

References

[1] Adams, R.;Sobolev Spaces, Academic Press, New York, 1975.

[2] Anane, A.,;Simplicite et isolation de la premiere valeur propre du p−Laplacien avec poids, Comptes Rendus Acad. Sc. Paris, 305, 725-728, 1987.

[3] Fleckinger, J.; Manasevich, R.; Stavrakakies, N.; De Thelin, F.; Principal Eigenvalues for some Quasilinear Elliptic Equations onR^N, Advances in Diff. Eqns., Vol. 2, No. 6, 981-1003, 1997.

[4] Francu, J;Solvability of Operator Equations, Survey Directed to Differential Equations, Lec- ture Notes of IMAMM 94, Proc. of the Seminar “Industerial Mathematics and Mathematical Modelling” , Rybnik, Univ. West Bohemia in Pilsen, Faculty of Applied Scinces, Dept. of Math., July, 4-8, 1994.

[5] Serag, H. and El-Zahrani, E.; Maximum Principle and Existence of Positive Solution for Nonlinear Systems on R^N, Electron. J. Diff. Eqns., Vol. 2005, No. 85, 1-12, 2005.

[6] Serag, H. and El-Zahrani, E.;Existence of Weak Solution for Nonlinear Elliptic Systems on R^N, Electron. J. Diff. Eqns., Vol. 2006, No. 69, 1-10,2006.

Salah A. Khafagy

Mathematics Department, Faculty of Science, Al-Azhar University, Nasr City 11884, Cairo, Egypt

E-mail address:el [email protected]

Hassan M. Serag

Mathematics Department, Faculty of Science, Al-Azhar University, Nasr City 11884, Cairo, Egypt

E-mail address:[email protected]